Let $A$ be a symmetric multilinear form on $\left(\mathbb{R}^d\right)^{\otimes n}\times \left(\mathbb{R}^d\right)^{\otimes n}$ and consider the random variable: \begin{align*} X=A(g_1,\ldots,g_n,g_1,\ldots,g_n) \end{align*} where $g_i\in \mathbb{R}^d$ are independent random vectors with i.i.d. random Gaussian entries of mean $0$ and variance $1$. Is there a closed form expression for $\mathbb{E}(X^2)$? In the case $n=1$ I know that $\mathbb{E}(X^2)=\|A\|_{F}^2$, where $\|\cdot\|_{F}$ is the Frobenius norm of the underlying norm. Does that generalize in any way for $n>1$? Thanks for any help!

For $n=1$, $$EX^2=(\text{tr}\,A)^2+2\|A\|_F^2, \tag{1} $$ not $\|A\|_F^2$.

Indeed, writing $A=(a_{ij})$ and $g_1=(Z_1,\dots,Z_d)$ with iid standard normal $Z_i$'s, we have $X=\sum_{i,j}a_{ij}Z_iZ_j$ and hence $$EX^2=\sum_{i,j,k,l}a_{ij}a_{kl}\,EZ_iZ_jZ_kZ_l. $$ Next, $EZ_iZ_jZ_kZ_l\ne0$ only if (i) $i=j\ne k=l$ (in which case $EZ_iZ_jZ_kZ_l=1$) or (ii) $i=k\ne j=l$ (in which case $EZ_iZ_jZ_kZ_l=1$) or (iii) $i=l\ne j=k$ (in which case $EZ_iZ_jZ_kZ_l=1$) or (iv) $i=j=k=l$ (in which case $EZ_iZ_jZ_kZ_l=3$). So, taking the symmetry of $A$ into account, we have $$EX^2=\sum_{i\ne k}a_{ii}a_{kk}+\sum_{i\ne j}a_{ij}^2+\sum_{i\ne j}a_{ij}a_{ji}+3\sum_i a_{ii}^2 \\ =\Big(\sum_i a_{ii}\Big)^2+2\sum_{i,j}a_{ij}^2, $$ so that (1) follows.

For $n>1$, $EX^2$ will be a big sum over all partitions of the set $\{1,\dots,4n\}$ into subsets of even cardinalities (cf. cases (i)--(iv) above).